axiomatic sets 16 (Peano's axioms)

By an axiom of infinity, the collection of all natural numbers was gotten.

By Peano's axioms, all natural numbers $\mathbb{N}$  has been defined.

$w$ which satisfies the following conditions (1)-(5) is $\mathbb{N}$ ;

(1)$0\in w$
(2)if any $n\in w$ , then $n\cup\left\{ n \right\}\in w$
(3)for each $n\in w$ , $n\cup\left\{ n \right\}\ne 0$
(4)if $x$  is a subset of $w$ such that $0\in x$  and if $n\in x$ , then $n\cup\left\{ n \right\}\in x$ ,
   then $x=w$
(5)if $n,m\in w$ and $n\cup\left\{ n \right\}=m\cup\left\{ m \right\}$ ,  then $n=m$

Please remember that
\[ n\cup\left\{ n \right\}=n+1 .   \]
By using preceding axioms, these can be proved.

$n\cup\left\{ n \right\}$  is called a successor set of $n$ .

Please note that (4) means the principle of "mathematical induction".
It may seem same as  an axiom of infinity.